II II II II II I - Waste Isolation Pilot Plant - U.S. Department of Energy

and assume that the plastic part of the strain rate is given by a normality condition
@ . ~
(4.5.9)
when the scalar multiplier,
y, must be determined.
A scalar measure of equivalent plastic strain rate is defined by
(4.5.10)
which is chosen
such that
(4.5.11)
The stress rate is assumed to be purely due to the elastic part of the strain rate and is expressed in terms of
Hooke’s law by
d=hrdeb+2pde1 (4.5.12)
where k and ~ are the Lame constants
for the material.
Below, we develop the theory for the cases of isotropic hardening, kinematic hardening, and combined
hardening separately so that the reader can see the details for each case.
4.5.1 Isotropic Hardening
In the isotropic hardening case, the backstress is zero and the stress difference is equal to the deviatoric stress, S.
We write a consistency condition by taking the rate of Equation (4.5.4)
f(@=2KK . (4.5.13)
By consistency we mean that the state of stress must remain on the yield surface at all times. We use the chain rule
and the definition of the normal to the yield surface given by Equation (4.5.7) to obtain
af . af
‘(o) ‘Xi:o = z ‘6
(4.5.14)
and from Equations (4.5.3) and (4.5.4)
(4.5.15)
Combining Equations (4.5. 13), (4.5. 14), and (4.5.15)
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